Three-dimensional X-ray imaging is based on taking several one-dimensional (1-D) or two-dimensional (2-D) projection images of a three-dimensional (3-D) body from different directions. If 1-D projection images are available from all around a 2-D slice of the body with dense angular sampling, the inner structure of the slice can be determined. This is known as Computerized Tomography (CT) imaging technology, which is widely used in medicine today. A crucial part of CT technology is the reconstruction algorithm taking the X-ray images as argument and returning a voxel representation of the 3-D body.
A collection of X-ray images of a 3-D body is called sparse projection data if (a) the images are taken from a limited angle of view or (b) there is only a small number of images. Sparse projection data does not contain sufficient information to completely describe the 3-D body. However, in many practical imaging situations only sparse projection data is available.
Traditional reconstruction algorithms such as filtered back-projection (FBP), Fourier reconstruction (FR) or algebraic reconstruction technique (ART) do not give satisfactory reconstructions from sparse projection data. Reasons for this include requirement for dense full-angle sampling of data and difficulty to use a priori information, for example non-negativity of the X-ray attenuation coefficient. In the case of limited-angle data, tomosynthesis can be applied to produce reconstructions of the body along 2-D slices through the body. However, the tomosynthetic slices suffer from blurring that severely compromises image quality.